Questions: Normal Distribution Introduction

Question 1

In a normal distribution with mean 70 and standard deviation 10, approximately what percentage of values fall between 60 and 80?

• A. 50%
• B. 68%
• C. 95%
• D. 99.7%

Answer: (B)

60 and 80 are each exactly 1 standard deviation from the mean (70 ± 10). The empirical rule states that approximately 68% of data in a normal distribution falls within 1 standard deviation of the mean.

Question 2

A z-score of 0 means the data value is 0.

• T. True
• F. False

Answer: False

A z-score of 0 means the data value equals the mean, not that the value itself is 0. The formula is z = (x − mean) / sd, so z = 0 precisely when x = mean, regardless of what the mean actually is.

Question 3

A dataset of human heights is approximately normal with mean 170 cm and standard deviation 8 cm. A height of 186 cm has a z-score of 2. What does that z-score tell you?

Model answer: It means 186 cm is 2 standard deviations above the mean. By the empirical rule, about 95% of heights fall within 2 standard deviations (154–186 cm), so 186 cm sits at the high end of the typical range.

z = (186 − 170) / 8 = 16 / 8 = 2. A z-score converts any value from any normal distribution into a universal scale: how many standard deviations above or below the mean. This allows meaningful comparisons across distributions with different means and spreads.